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Inequalities for the curvature of curves and surfaces. (English) Zbl 1161.53002

This article discusses bounds for the total mean curvature of two closed surfaces in 3-space of same topology in terms of their total absolute curvatures and their Fréchet distance. This result also applies to bound the difference of the lengths of two curves. Using methods from algebraic topology the difference of the total mean curvatures of two surfaces can be obtained without bounds for the angles of their normals. This result applies also to estimate the total mean curvature of a smooth surface from a piecewise-linear approximation.

MSC:

53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces
53C65 Integral geometry
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